THESE DE DOCTORAT DE
L’UNIVERSITE PIERRE ET MARIE CURIE
SpécialitéChimie et Physico-Chimie des Polymères (ED 397, Physique et Chimie des Matériaux)
Présentée par
Etienne DUCROT
Pour obtenir le grade de
DOCTEUR de l’UNIVERSITÉ PIERRE ET MARIE CURIE
Sujet de la thèse :
Innovative tough elastomers:
Designed sacrificial bonds in multiple networks
Soutenue le 11 Décembre 2013 devant le jury composé de :
M. Tristan BAUMBERGER Examinateur
M. Laurent BOUTEILLER Examinateur
M. Hugh BROWN Examinateur
M. Markus BULTERS Exalminateur
M. Costantino CRETON Directeur de thèse
Mme. Jian Ping GONG Rapporteur
REMERCIEMENTS
Je souhaite tout d’abord remercier Christian Frétigny, directeur du Laboratoire PPMD-SIMM de l’ESPCI ParisTech, de m’avoir accueilli et fourni tous les éléments nécessaires pour effectuer ma thèse.
Je tiens à remercier chaleureusement mon directeur de thèse, Costantino Creton qui m’a offert ce beau sujet de thèse et qui m’a accompagné pendant ces trois années. Merci pour la très grande liberté et la confiance qu’il a su me donner. J’ai beaucoup appris de nos nombreuses discussions et de ses précieux conseils. Grâce à lui ma thèse a été ponctuée par de nombreuses conférences aussi bien en France à Lyon et Anduze qu’à l’étranger, Rolduc, Jackson Hole, Pise et Rome. Et ce même jusqu’à la semaine précédent la soutenance, lors de cette dernière conférence à Boston. Ce travail est le fruit d’une collaboration avec DSM, je leur suis grandement reconnaissant pour le financement de cette thèse. Je remercie particulièrement Markus Bulters, Paul Steeman, Meredith Wisemann et Carel Fitié pour avoir porté un grand intérêt à ce projet, pour nos réunions enrichissantes et stimulantes, nous poussant toujours plus loin dans la compréhension du système.
Mes remerciements vont également aux membres du jury pout l’intérêt qu’ils ont témoigné à l’égard de mon travail. Merci particulièrement à Jian Ping Gong et Julian Oberdisse d’avoir accepté de rapporter ce manuscrit. Merci également à Laurent Bouteiller de m’avoir accueilli dans son laboratoire au début de la thèse pour m’initier à la polymérisation contrôlée et pour avoir présidé ce jury. Enfin je remercie Hugh Brown et Tristan Baumberger d’avoir participé à ce jury et apporté un regard critique à mon travail.
Lors de cette thèse j’ai eu la grande chance de collaborer avec de nombreuses équipes hors de l’ESPCI, ouvrant le spectre des techniques mises en œuvre. J’exprime ma grande gratitude envers Rint Sijbesma et Yulan Chen de TU/e à Eindhoven pour la synthèse du réticulant mechanoluminescent, qui une fois incorporé dans nos échantillons a mené à une preuve visuelle des mécanismes moléculaires mis en jeu.
Arno Kentgens et Wanling Shen de l’université de Nijmegen ont débuté des analyses de RMN du solide sur nos échantillons, les résultats sont encore en cours d’analyse mais je tiens à relever leur travail.
Je remercie Sylvie Castagnet et Ousseynou Kane-Diallo de l’ENSMA à Poitiers pour m’avoir si bien accueilli au sein de leur laboratoire et initié aux joies de la cavitation sous pression de gaz. Je tiens à remercier Fabrice Cousin, François Boué, Arnaud Hélary et Alain Lapp du Laboratoire Léon Brillouin (CEA, Saclay) pour leur accueil amical lors de mes nombreuses sessions sur Orphée pour la diffusion de neutrons aux petits angles. La neige a également été une compagne régulière lors de ces visites me retenant sur le plateau plus que de raison. Il est fort plaisant d’aller régler le spectro en plein milieu de la nuit et de surprendre les lapins dans la neige !
Le travail expérimental n’aurait pas eu la même réussite sans l’aide précieuse apportée par David Martina pour la conception, la réalisation et la mise en place de montages expérimentaux. Nos discussions parfois animées ont bien souvent transformé de simples ébauches en des montages performants. Merci beaucoup également à Ludovic Olanier pour la fabrication des pièces sur mesure.
J’aimerais remercier tout particulièrement Hélène Montes pour les chouquettes de rédaction, sa bonne humeur communicative et sa bienveillance depuis de nombreuses années. Merci aussi d’avoir été d’une aide précieuse dans l’analyse des données de diffusion, mettant le coup final à l’analyse après des mois de tentatives infructueuses.
J’en profite pour souhaiter de très beaux résultats à Thitima et Pierre qui commencent tout juste leur thèses et prennent en main la suite du projet.
J’ai eu la chance d’effectuer ma thèse dans un laboratoire où les échanges sont très faciles avec les permanents et les non permanents, je leur adresse ici mes sincères remerciements. Merci à mes co-bureau David, François, Koichi et Benjamin, à la « girl team » d’en face Séverine, Clémence et Jennifer et aux nombreux voisins : Aurélie, Basile, Céline, Eloïse, Eric, Guillaume, Lucie, Natasha, Nisita, Peiluo, Quentin, Rémi, Robin, Sandrine, Solenn, Xavier, Yannick et tous ceux avec qui j’ai partagé de très bons moments durant ces années, des bières, des vacances, des week-end de rédaction ...
Merci à ceux qui font que le laboratoire tourne : Freddy, Mohamed, Gilles, Pierre, Flore et Annie. Et merci à Laurence, alias Jacqueline Vabre, la plus charmante marchande de café à l’accent qui chante !
Au cours de ma thèse, j’ai eu l’opportunité d’enseigner à l’ESPCI dans le cadre des tutorats. Je tiens à remercier les enseignants qui m’ont fait confiance : Jean-Louis Halary, Hélène Montes, Sophie Norvez et Corinne Soulié.
Je remercie mes amis et tous ceux qui ont marqué ces années de thèse et qui n’ont pas été nommés jusque là. Une pensée particulière à mes deux colocs Aurélie et Charlotte pour tous les très bons moments passés ensemble !
J’adresse enfin mes plus vifs remerciements à mes parents et à ma sœur pour leur soutien sans faille et leurs encouragements depuis si longtemps.
AFM Atomic Force miscroscopy
ARTC Atom Transfer Radical Coupling
ATRP Atom Transfer Radical Polymerization
BA Butyl Acrylate
BADB Bis(adamantly)-1,2-dioxetane bisacrylate
BDA 1,4-Butandiol diacrylate
BHT Butylated hydroxytoluene
DBU 1,8-Diazabicyclo[5.4.0]undec-7-ene DCC N,N’-Dicyclohexylcarbodiimide
DMA Dynamic Mechanical Analysis
DMPA 4-(Dimethylamino) pyridine
DMF N,N-Dimethylformamide
DN Double Network
EA Ethyl Acrylate
EAD Deuterated Ethyl Acrylate
EBBIB Ethylene bis(2-bromoisobutyrate)
HMP 2-Hydroxyethyl-2-methylpriopiophenone
IPN Interpenetrated Polymer Network
MA Methyl Acrylate
MCH 6-Mercapto-1-hexanol
NMR Nuclear Magnetic Resonance
OctSn Tin(II) 2-ethylhexanoate
PAAm Poly(acrylamide)
PAMPS Poly(2-acrylamido-2-methylpropanesulfonic acid)
PBA Poly(butyl acrulate)
PDMS Poly(dimethylsiloxane)
PEA Poly(ethyl acrylate)
PEG Poly(ethylene glycol)
PET Poly(ethylene terephthalate)
PMA Poly(methyl acrylate)
PMDETA N,N,N’,N’’,N’’-Pentamethyl-diethylenetriamine
PMMA Poly(methyl methacrylate)
SANS Small Angle Neutron Scattering
sCMOS Scientific Complementary metal-oxide semiconductor
SN Single or Simple Network
THF Tetrahydrofuran
TN Triple Network
L
IST OFS
YMBOLSE Elastic modulus
E’ Storage modulus
E’’ Loss modulus
Chain Stiffness, Characteristic Ratio
Fracture toughness
I(q) Scattering intensity
kB Boltzmann constant
Me Average weight between entanglements
Mx Average weight between crosslinks
ppm Part per million
q Scattering wave vector
Q Swelling ration in volumr
R Gas constant
T Temperature
Tg Glass transition temperature
Strain
Λ Wavelength
Weight fraction
ρ Density
ρ Scattering length density
Nominal stress
Mooney stress
Nominal stress
True stress
General Introduction
... 1Chapter 1
–Physics of polymer networks and networks design ... 5Introduction ... 7
1- General concepts on rubber elasticity ... 8
1-1- Ideal chain model ... 8
1-2- Entropy and free energy of an ideal chain ... 11
1-3- Polymer networks ... 12
1-3-1- Affine network model ... 12
1-4- Crosslinks and entanglements ... 14
1-4-1- At small strain ... 15
1-4-2- At intermediate strain... 16
Mooney-Rivlin model ... 16
Rubinstein and Panyukov model ... 17
1-5- High strain region ... 18
2- Fracture mechanics of rubbers ... 20
2-1- Model of Lake and Thomas ... 20
2-2- Experimental evaluation of the fracture toughness ... 23
3- Network design ... 25
3-1- Bimodal networks ... 25
3-2- Interpenetrated networks ... 27
3-2-1- Double networks hydrogels ... 28
Synthesis and structure ... 28
Mechanical properties and toughening mechanism ... 28
Model for the fracture of DN hydrogels ... 30
Recent developments on DN gel structure ... 32
3-2-2- Interpenetrated elastomers ... 33
3-3- Introduction of prestretched chains in a polymer network ... 35
Conclusions and objectives of the manuscript ... 37
1- Synthesis ... 46
1-1- Chemicals ... 46
1-2- Polymerization Conditions and Environment ... 47
1-3- General path to multiple networks elastomers ... 49
1-3-1- First polymerization (Simple networks) ... 49
Samples compositions ... 49
Solvent for deswelling ... 50
Shrinking and digitations ... 50
1-3-2- Second polymerization (Double networks) ... 51
Samples compositions ... 52
1-3-3- Third polymerization (Triple networks) ... 53
1-4- Variations on the second network ... 55
1-5- Solvent free simple networks ... 56
1-5-1- Second/Third Network alone ... 56
1-5-2- Simple networks of EA ... 56
2- Extractable and conversion ... 57
2-1- First networks ... 57
2-2- Bulk samples ... 57
3- Structural properties by thermomechanical analysis ... 58
3-1- Theoretical background on DMA ... 58
3-2- Results and discussion ... 60
3-2-1- Simple networks ... 61
3-2-2- Multiple networks with a single monomer type ... 61
3-2-3- Multiple networks with a contrast in monomer ... 62
Miscible Multiple networks ... 63
Immiscible Multiple networks ... 63
Conclusions ... 65
1- Material and methods ... 70
1-1- Mechanical testing experiments ... 70
1-2- Tensile tests ... 71
1-3- Step-Cycle extension ... 72
1-4- Fracture in single edge notch test ... 73
2- Simple networks alone: weak elastomers ... 73
2-1- Effect of the crosslinker concentration in simple networks ... 73
2-2- First network: brittle and tunable ... 78
Crosslinker concentration ... 78
Nature of the monomer ... 80
2-3- Fracture properties of simple networks ... 82
3- Multiple networks: stretching the prestretched ... 83
3-1- General behavior ... 83
3-2- Origin of the initial modulus ... 86
3-3- The origin of stiffening and softening ... 89
3-4- Less extensible second network ... 94
4- Variations of monomers ... 95
4-1- Changes in the second network ... 95
4-2- Changes in the composition of the first network ... 97
5- Cyclic extension: from pure elasticity to bulk dissipation ... 99
5-1- Viscoelastic behavior in second networks alone ... 99
5-2- Perfect reversibility of the elasticity in double networks ... 100
5-3- Mullins effect in Triple networks ... 101
5-4- Energy dissipation and bond breaking mechanism ... 103
6- Fracture properties ... 106
6-1- Experimental results and discussion ... 106
6-2- Models ... 110
Lake and Thomas ... 110
Brown and Tanaka models for DN hydrogels ... 112
Conclusion ... 115
1- Small Angle Neutron Scattering to probe polymeric chains conformations... 122
1-1- Generalities ... 122
1-2- Scattering from polymer melts ... 124
1-3- Scattering and anisotropy ... 125
1-4- Experimental setup ... 129
1-5- Data treatment ... 130
2- Labeled multiple networks elastomers: from monomer synthesis to mechanical properties ... 132
2-1- Synthesis of deuterated monomers ... 133
2-1-1- Procedure ... 134
2-1-2- Analysis ... 135
2-2- Labeled multiple networks ... 135
2-3- Check of the mechanical properties ... 137
3- Chains of the first network: undeformed samples ... 138
3-1- Partially deuterated multiple networks of pure poly(ethyl acrylate) ... 138
3-2- Comparison between EA and MA as second/third monomers ... 142
4- Stretching the prestretched chains ... 143
4-1- Scattering pattern of stretched multiple networks ... 143
4-2- Scattering along the principal directions ... 145
4-2-1- In double networks ... 145
4-2-2- In triple networks ... 147
4-2-3- Sacrificial bonds and scattering ... 148
Summary of SANS results ... 150
References ... 151
Chapter 5
– Molecular toughening mechanism, mechanoluminescence as a probe for bond breaking ... 153Introduction ... 155
1- Colors and light in polymers under stress ... 156
2- Mechanoluminescent crosslinker in multiple network elastomers ... 157
2-1- Mechanoluminescent crosslinker ... 157
Successful incorporation in networks ... 160
3- Dioxetane as a probe of a bond breaking mechanism ... 160
3-1- Experimental conditions and data treatment ... 160
3-2- Results ... 163
3-2-1- Mechanical properties ... 163
3-2-2- Luminescence signal ... 164
3-3- Luminescence and mechanical properties ... 166
3-4- Stress or Strain sensor? ... 168
4- Fracture and luminescence ... 170
4-1- Experimental conditions and data treatment ... 170
4-1-1- Mechanical part of the experiment ... 170
4-1-2- Optical part of the experiment ... 171
4-2- Results ... 172
4-2-1- Fracture toughness ... 172
4-2-2- Before crack propagation ... 173
4-2-3- Crack propagation ... 174
Simple network EA0.5m ... 174
Double and Triple networks ... 175
Quantitative analysis ... 176
Crack velocity ... 178
4-3- Molecular description of the dissipation mechanism ... 179
Conclusion ... 182
References ... 183
Chapter 6
– From heterogeneous to Homogeneous first network ... 185Introduction ... 187
1- Homogeneous first network in multiple networks elastomers ... 188
1-1- Chain synthesis by Atom Transfer Radical Polymerization ... 188
1-1-1- ATRP Principle ... 188
1-1-2- Initiator synthesis ... 189
Protocol ... 195
1-3- Perfect first network ... 197
Protocol ... 198
1-4- Synthesis of DN and TN ... 199
1-5- Comments on other methods to end-functionalize the chains ... 199
Atom Transfer Radical Coupling ... 199
Click chemistry: alkyne /azide reaction ... 201
2- Properties of ‘perfect’ multiple networks ... 202
2-1- Not so ‘Perfect’ first network ... 202
2-2- ‘Perfect’ multiple networks under deformation ... 203
2-1- Fracture of ‘perfect’ multiple networks ... 207
Conclusion ... 208
References ... 209
General Conclusion and Outlook
... 211Résumé Substantiel en français
... 215List of references
... 229Annexes
... iAnnex 1: Mechanoluminescent crosslinker: polymerization kinetics and thermal stability ... iii
Elastomers are made of crosslinked flexible and highly entangled polymers with a low glass transition temperature. They are widely used because of their large reversible deformability up to strains of several hundred percent. However, unfilled elastomers suffer from a tradeoff between fracture toughness and stiffness. To stiffen a pure polymer network, the first idea is to increase the crosslinker concentration which leads to a more brittle material. This behavior has been well described and modeled by Lake and Thomas in the late sixties and is still the background for the understanding of fracture toughness. They perfectly described the threshold value of the fracture toughness at small strain rate or high temperature which is low for every polymer network and only depends on the number of chemical bonds between crosslinks and the dissociative energy of a bond.
To circumvent this limitation, the rubber industry widely used inorganic nanofillers (carbon black, silica…) to toughen materials. Fillers particles were found to cause highly dissipative process due to molecular friction that increases both toughness and stiffness of filled materials. But there are limitations, and especially at high temperature, where the fracture toughness effectively decreases until the threshold value of an unfilled network. Some high tech or bio applications cannot afford the presence of fillers, or need high chemical or environmental resistance (temperature, UV…) and inertness. Polymers that can fulfill those specifications are usually loosely entangled and exhibit very weak mechanical properties.
Many strategies have been tried to increase the toughness and the stiffness elastomers without the use of fillers, most of them using a network design strategy. In multimodal networks, the idea was to introduce controlled heterogeneities in the network, short chains crosslinked with long chains. But results were not as impressive as expected and they only showed a moderate improvement of the fracture toughness. Another strategy was to stretch a partially crosslinked network and to perform a second crosslinking reaction under stretch. The fracture toughness was improved in the prestretching direction but materials were highly anisotropic and show a decrease in stiffness at small strain.
New ideas on network design to reinforce soft and brittle materials have emerged over the past ten years, and especially on the field of hydrogels. Those highly swollen networks are usually or stiff and brittle or extremely soft and very extensible. Recent work reported levels of toughness in excess of 100 to 1000 times of the regular gels while maintaining reasonable extensibility and a
recoverability of the strain and with no help from viscoelasticity. The increase in toughness has been attributed to an internal bond breaking mechanism of weak or overstressed bond distributed in the entire material and designed to break before macroscopic failure of the material. Those sacrificial bonds may be hydrogen bonds between polymer chains or carbon-carbon bonds on highly extended polyelectrolyte chains in an interpenetrated polymer networks structure.
Inspired from sacrificial bonds in nature and from published recent work on hydrogels, we aim to design innovative interpenetrated network elastomers that contain isotropically stretched chains and show a selective bond breaking mechanism in the bulk to dissipate energy before fracture. Using interpenetrated networks to toughen elastomer has been tried before but with not much success and poses significant challenges. The presence of water in hydrogels favors mixing of different polymers. It is also a great solution to get highly extended polymer chains by using polyelectrolytes. In the dry state, polymer mixing is often more difficult due to polymer interactions. A high degree of swelling is also difficult to obtain with hydrophobic polymers in organic solvents. However, in the dry state, polymers are more entangled than in hydrogels and may help to toughen the material.
This manuscript is divided into six chapters. The first one is devoted to important background information on the physics and chemistry of rubbers and to a review of published papers on network designs in hydrogels and elastomer networks. The purpose is to gather elements essential to the design of innovative tough elastomers and to understand their properties.
The second chapter is dedicated to the synthesis and small strain characterization of simple and multiple networks elastomers. The large strain properties and fracture toughnesses are presented and commented on Chapter 3 where hypotheses are made on the molecular mechanism responsible for the stiffening and toughening of multiple networks.
In Chapter 4, the structure of the first network in multiple networks is investigated and analyzed with small angle neutron scattering experiments. A custom-made device was used to stretch the sample in the neutron beam and follow the conformation of chains under mechanical stress.
In Chapter 5, the molecular mechanism hypothesized to explain the toughening of multiple networks is tested with the incorporation of a novel mechanoluminescent crosslinker in multiple networks.
In Chapter 6, the influence of the structure of the first network is studied by comparing standard random first networks with a model first network obtained from end-funtionalized chains. The impact of its incorporation in multiple networks on their mechanical properties and fracture toughness is then investigated.
Finally, the main contributions of this work are summarized in a concluding part along with outlooks.
Chapter 1 –Physics of polymer networks and networks design ... 5
Introduction ... 7
1- General concepts on rubber elasticity ... 8
1-1- Ideal chain model ... 8
1-2- Entropy and free energy of an ideal chain ... 11
1-3- Polymer networks ... 12
1-3-1- Affine network model ... 12
1-4- Crosslinks and entanglements ... 14
1-4-1- At small strain ... 15
1-4-2- At intermediate strain... 16
Mooney-Rivlin model ... 16
Rubinstein and Panyukov model ... 17
1-5- High strain region ... 18
2- Fracture mechanics of rubbers ... 20
2-1- Model of Lake and Thomas ... 21
2-2- Experimental evaluation of the fracture toughness ... 23
3- Network design ... 25
3-1- Bimodal networks ... 25
3-2- Interpenetrated networks ... 27
3-2-1- Double networks hydrogels ... 28
Synthesis and structure ... 28
Mechanical properties and toughening mechanism ... 28
Model for the fracture of DN hydrogels ... 30
Recent developments on DN gel structure ... 32
3-2-2- Interpenetrated elastomers ... 33
3-3- Introduction of prestretched chains in a polymer network ... 35
Conclusions and objectives of the manuscript ... 37
I
NTRODUCTIONThe main purpose of this work is the design of innovative elastomeric materials composed of pure polymer with no use of fillers and to study their structure/properties relationships. There is an abundant literature available on polymer networks and strategies to toughen them by varying the architecture of the network. The field of polymer gels in particular has been very prolific during the last decade and is a source of inspiration for the creation of novel concepts applicable to elastomers.
In this introductive chapter we first recall some general notions on the physics and chemistry of rubbers. We start with the description of the ideal chain, continue with the classical rubber elasticity theory and extend it to the behavior of polymer network at large strains.
In a second part we introduce the notion of fracture mechanics in rubbery materials based on molecular considerations and the experimental determination of the fracture toughness of a material.
Then in a brief literature survey we examine some strategies implemented to toughen networks via a design at the molecular level. We report previous studies on elastomers as well as strategies recently developed to improve the mechanical properties of polymer gels that are usually very brittle materials.
Finally, the objectives of the thesis and the general approach which has been followed will be described.
1- General concepts on rubber elasticity
A material that is loaded with an external force is generally deformed. If the material recovers instantly its original dimensions when the applied force is removed, then the deformation is said to be elastic and reversible. A large majority of solids have a limit of deformation, below which, the elasticity is linear.
The ideal behavior of an elastic material is independent of time. As a response to a stress σ (which is the ratio between the applied force and the application area F/S), it deforms proportionally regardless of the rate at which the stress is applied. In the case of uniaxial deformation, the stress and the strain are simply linked by Hooke’s law:
Eq. 1 where is the relative strain. The constant of proportionality E is the Young’s modulus.
Materials that follow this law are called Hookeans and the maximal of extensibility, for a given material, that define the limit of applicability of Hooke’s law is called limit of elasticity.
Generally polymers follow this linear elasticity only at short observation times and small deformations. Crosslinked polymers at temperatures higher than their glass transition temperature (Tg) are elastic over much larger strain ranges due to entropic elasticity, a central
concept in mechanical properties of rubber.
1-1- Ideal chain model
We recall here the main results on the conformation of an ideal chain, i.e. with no interactions between monomers or between chains and solvent. This model is the starting point for many models used in polymer physics1,2.
We start with a flexible chain made of N+1 atoms Ai (0 ≤ i ≤ N) defining N segments with a
length . Each segment is described by a vector , oriented randomly. The end-to-end vector of the chain is defined as the sum of :
Figure 1: Schematic of an ideal chain of +1 monomers, and of its end-to-end vector
We may note that for a set of a large number of chains, the mean value of the end-to-end vector is zero, a consequence of the randomness of directions.
0 Eq. 3
The mean-square end-to-end distance of a chain is given by Eq. 4 and the corresponding end-to-end distance by Eq. 5.
Eq. 4
⁄ ⁄ Eq. 5
This distance is considerably shorter than the fully extended chain, which is equal to N . This result indicates that the chain is curled around in a relatively compact way and it is designated as a random coil.
Around the average value given by Eq. 4, the distribution of end-to-end distances is Gaussian. The probability of finding a chain with an end-to-end distance between R and R+dR, within a sample, is expressed as Eq. 6.
, 3
2
⁄
e 4 Eq. 6
In a more realistic polymer chain, there are correlations between over a given range. Two segments far from each other may nevertheless not be correlated. Eq. 4 must be corrected by the introduction of a parameter called the characteristic ratio that depends on the number of links and that represents the flexibility or the rigidity of the chain. In the case of an ideal chain
Eq. 7
When describing a real flexible chain, it can be interesting to consider an equivalent ideal chain with no correlation between monomers. Kuhn describes the conformation of a real chain made of monomers of size by an equivalent ideal chains composed of segments of size k
freely joined. Then the mean-square end-to-end distance, is given by Eq. 8 for a chain under no deformation.
Figure 2: Schematic representation of an equivalent Kuhn chain
The totally extended end-to-end distance, for such a chain is given by Eq. 9.
The maximum extensibility of a Kuhn chain corresponds to the ratio between and and is simply linked with (Eq. 10).
To conclude on the notion of Kuhn segment, we report the link between Kuhn parameters, and , and real characteristics of the chain, and ℓ the number and length of C-C bonds and the angle between two neighbouring bonds ( 68 °). Expressions of the end-to-end length and its maximal value in case of total extension of the chain are given on Eq. 11 and Eq. 12. The expressions of and are easily extracted (Eq. 13 and Eq. 14).
Eq. 8
Eq. 9
⁄ ⁄
⁄
Figure 3: Scheme of a chain in the all-trans conformation
1-2- Entropy and free energy of an ideal chain
Entropy is the product of the Boltzmann constant by the logarithm of the number of states Ω. Here Ω is the number of conformations of a freely jointed chain of monomers with end-to-end vector . Entropy is then a function of N and .
Applying the probability distribution function of an ideal chain, the entropy is given by Eq. 16.
where , 0 is a term which only depends on . The free energy of the system is defined as:
The energy of an ideal chain is independent on the end-to-end vector , since the monomers of the ideal chain have no interaction energy. The free energy can be written:
ℓ Eq. 11 ℓ cos 2 Eq. 12 cos 2 Eq. 13 ℓ cos 2 Eq. 14 , ln Ω , Eq. 15 , 3 2 . , 0 Eq. 16 T Eq. 17
1-3- Polymer networks
We now focus on a network of ideal chains, connected randomly by crosslinks.
In the case of a polymeric network, the first law of thermodynamics states that the changes in internal energy of the system is the sum of all energy changes (heat added to the system, work done to change the network volume and work done upon network deformation). This leads to the following expression for the free energy:
It is important to note that the applied force f to deform the network consists on two contributions:
with fE the internal energy term, and fS the entropic term
In conventional crystals, such as metals, the energetic contribution is dominant, distances between atoms change from their equilibrium position, increasing the internal energy of the system. In rubbers, the entropic contribution to the force is more important and governs the force. For an ideal network, fE 0. From now on, this contribution will be neglected.
1-3-1- Affine network model
We consider a polymer network only composed of crosslinked polymer chains with an average molecular weight between crosslinks x smaller than the average molecular weight between
entanglements e.
The high deformability of such a network arises from the entropic elasticity of the polymer chains that make up the network. The simplest model that captures this idea of rubber elasticity is the affine network model originally proposed by Kuhn.
The main assumption of the affine deformation model is that the average deformation of the elastic strands is identical with the macroscopic applied deformation.
We consider a network of initial dimensions Lx0, Ly0, Lz0. If the network is deformed in the three
directions by the factors λx, λy and λz, then the dimensions of the deformed networks are given by
Eq. 21. Lx = λxLx0 Ly = λyLy0 Lz= λzLz0 Eq. 21 d dT pdV fdL Eq. 19 f ∂ ∂L V,T ∂ T ∂L V,T ∂ ∂L V,T T ∂ ∂L V,T fE fS Eq. 20
Assuming that there is no volume change of the material under deformation, which is in general a verified hypothesis, values of λ in the three directions are linked by Eq. 22.
λ λ λ 1 Eq. 22
Assuming an elastic strand composed of monomer, the end to end vector of the chain is defined as in its initial state, with Rx0, Ry0, Rz0, the projections on the three directions of the
plane. Accordingly, the end-to-end of the vector in the deformed state is given by Eq. 23.
Rx = λxRx0 Ry = λyRy0 Rz= λzRz0 Eq. 23
Considering thermodynamics on a network made of crosslinked ideal chains composed of monomers, deformed in constant volume conditions, the variation of entropy of the network is given by Eq. 24.
Δ
2 3 Eq. 24
The free energy of the system is also given by Eq. 25.
Δ Δ
2 3 Eq. 25
We now consider a uniaxial deformation along the x axis. Assuming conservation of volume 1 , the deformation can be defined only by the stretch λ (Eq. 26).
λ λ 1
λ λ 1
√λ
Eq. 26 The free energy of the system can now be written as follows in Eq. 27.
Δ
2
2
3 Eq. 27
The force required to deform the network in the x direction is related to the free energy and then to λ by Eq. 28. f ∂Δ ∂L 1 ∂Δ ∂λ 1 Eq. 28 We define now the true stress σT as the ratio between the force fx and the transversal deformed
section LyLz. Then in the case of a uniaxial deformation along the x axis: σT
f 1 1 1
Eq. 29 We may also define the nominal stress σ as the ratio between the force and the initial cross
σN f Eq. 30
The constant coefficient linking the stress to the strain can be related to the Young’s modulus E according to Eq. 31.
E 3 3 3 Eq. 31
with ν = /V, the number of network strands per unit volume, the network density, Ms the
average molar mass of a network strand and R the gas constant.
The network modulus increases linearly with the temperature due to its entropic origin. It also increases linearly with the density of elastic strands. The equation shows that the modulus of a network is given by per elastic chain.
The affine prediction for both true stress and nominal stress in uniaxial deformation at constant network volume can be rewritten using the Young’s modulus:
σT E 3 1 Eq. 32 σN E 3 λ 1 Eq. 33 This relation is generally in good agreement with experimental data in the small strain regime, such materials are called neo-Hookean (in the small strain region, 1, the Hooke’s law is recovered, then the stress becomes non linear with the strain).
In a real network, one can also take into account defects, such as pendant chains (chains attached only at one end), polymer loops (intramolecular reactions) which are elastically inactive chains. The amount of inactive chains of every type is difficult to measure experimentally. In networks with long strands between crosslinks, chains form entanglements that may contribute to the mechanical properties at small and large strains may be considered.
1-4- Crosslinks and entanglements
The presented model of the deformation of ideal networks does not explain why the modulus of a network of extremely long strands does not decrease continuously toward zero but reaches a plateau. In models already mentioned, apart from the strand ends, monomers do not feel any constraining potential. In a real network made of long and linear chains, network chains impose
topological constraints on each other because they cannot cross. Edwards showed that the topological interactions of neighboring chains restrict the transverse fluctuations of a network strand to a confining tube of diameter .
This tube diameter can be interpreted as the end-to-end distance of an entanglement strand of monomers, with a molecular weight M such as:
⁄ Eq. 34
Entanglements are dynamic topological constraints that can relax with time and in which chains may slide when stretched. At small strain and observation time shorter than the terminal relaxation time, they might be seen as permanent crosslinks, before sliding and revealing their non permanent nature. Typically for poly(methyl acrylate) M = 11000 g.mol-1, 13000 g.mol-1 for
poly(ethyl acrylate) and 26000 g.mol-1 for poly(butyl acrylate)3.
Figure 4: Schematic representation of a real network
1-4-1- At small strain
The eventual entanglements must now be then taken into account for the prediction of the modulus and become the controlling parameter for loosely crosslinked networks, so with very long elastic strands. In such cases, the Young’s modulus can be decomposed in the contribution stemming from crosslinks Ex and from entanglements Ee.
E E E E 3ρRT M and E 3ρRT M Eq. 35 with Me the weight between entanglements and Mx the weight between crosslinks.
The modulus is controlled by crosslinks for low molecular mass strands between crosslinks (E Ex for Mx < Me) and by entanglements for high molar mass between crosslinks (E Ee for Mx > Me).
The deformation dependence of the stress in the Edwards tube model is the same as in the classical models because each entanglement effectively acts as another crosslink junction in the network. Therefore, the Edwards tube model is unable to explain the strain softening at intermediate deformation. The reason for the classical functional form of the stress/strain dependence is that the confining potential is assumed to be independent of deformation.
1-4-2- At intermediate strain
Mooney-Rivlin model
The model of Mooney-Rivlin4 is an alternative to molecular phenomenological models presented
until now. The behavior of rubbers is indeed only partially described by the previous models that are mainly relevant at small strain.
Mooney and Rivlin proposed a general expression of the free energy with no molecular interpretation of the terms.
The main hypotheses of their model are that the elastomer is incompressible and isotropic in its undeformed state.
The Mooney Rivlin model starts from the three invariants of the deformation: I
I I
Eq. 36
The free energy density of the network /V can be written as power series in the difference of these invariants from their values in the undeformed network 1 :
V 3 3 1 Eq. 37
The second term in the series is analogous to the free energy of the classical models
3 3 Eq. 38
The third term describes the deviations from the classical neo-Hookean behavior, and the fourth term is null due to volume conservation and incompressibility.
For uniaxial deformation of an incompressible network, the Mooney-Rivlin free energy density restricted to the first order is written as:
V
2
3 2 1 3 Eq. 39
The true stress can be obtained from the free energy density:
σT 1 2 2 1 Eq. 40
With the same method, the nominal stress can also be calculated and we can also introduce the Mooney stress σM .
σM 1 1 2 2 Eq. 41
The classical Mooney-Rivlin representation for rubbers is the plot of the Mooney stress as a function of the inverse of λ. A direct observation of the plot gives information on the difference between the behavior of the real network and the classical rubber elasticity 0 when the curve is horizontal. The presence of entanglements is detected by a softening in uniaxial tension 0 as the stretch increases. The presence of prestretched chains in the network can result in a stiffening 0 . This empirical model becomes however completely incorrect in the case of uniaxial compression or biaxial tension where C2 > 0 results in a stiffening of the material
which is not observed experimentally.
Rubinstein and Panyukov model
To complete the Edwards tube model and extend it to higher deformations, Rubinstein and Panyukov5,6 proposed a non affine tube model in which the randomness of the crosslinking
process is taken into account as well as the deformation of the tube when the sample is stretched. This approach leads to a relation between the Mooney stress and modulus from entanglements and crosslinks, E and E respectively.
This solution is in good agreement with experiments on uniaxial deformation of networks in tension but still overpredicts the stress required to compress a network. Rubinstein and Panyukov introduced the non-affine slip-tube model taking into account that chains along the deformation are elongated and compressed on others. Stored length from the compressed directions of the tube can redistribute itself into the stretched directions, balancing the tension in all directions and lowering the free energy and the stress in the network. The resulting dependence of stress on the deformation in the non-affine slip-tube model does not have a simple analytical form. However, the model has been solved numerically and its solution in the experimentally relevant range of 0.1 < λ < 10 can be approximated in a form similar to Eq. 43.
Eq. 43 can be reduced to Eq. 35 in the small deformation limit (λ → 1). This simple equation separates the contribution from entanglements from that of crosslinks and hence allows them to be determined experimentally.
1-5- High strain region
It is important to note that experimentally, a lot of elastomers show important stiffening at high strain as presented in Figure 5.
Figure 5: Comparison between experiemnt and gaussien theory for a sample under uniaxial deformation
1 ⁄ 1 Eq. 42
1 1
The softening in the intermediate strain region has already been explained by the behavior of entanglements. We now concentrate on the properties at higher elongation than λ = 6 in this example. The stiffening observed experimentally is not explained by the Gaussian theory or by the model of Rubinstein and Panyukov. It is the result of a non-Gaussian statistics for highly stretched polymer chains. The Gaussian approximation used in those models does not inject any limitation in the elongation, it is only valid for end-to-end distances well below the fully extended state.
The energy needed to stretch a chain diverges at Rmax, due to the finite extensibility of chains.
Highly stretched chains are no more in random coil conformations but oriented along the loading direction.
This effect can be computed by the theory of non-Gaussian network and the introduction of the Langevin statistics to model the finite extensibility of chains. Arruda and Boyce7 developed a
phenomenological model (Eq. 44) based on the description of eight chains assuming an affine network model.
An approximation of the Langevin function leads to an approximate expression of the nominal stress Eq. 45 for a network under uniaxial deformation.
As mentioned earlier, the affine network model is not the most appropriate model for networks crosslinked and entangled2. Therefore, the more accurate model of Mooney-Rivlin can be
injected to get a better description of the system in Eq. 46.
with the number of Kuhn segments per elastic strands, and , the Mooney-Rivlin parameters. 9 . Eq. 44 1 3 1 Eq. 45 2 1 1 3 1 Eq. 46
The model proposed by Gent8 may also be cited here, as it also introduces the finite extensibility
of polymer chains in a particularly simple way. This model links the nominal stress in uniaxial extension to the stretch through the simple empirical formula.
with E and the two adjustable parameters, the latter being related to λm the finite extensibility
of the network by Eq. 49.
Like the Arruda-Boyce model, The Gent model can be used to fit experimental curves both on compression and uniaxial extension as long as entanglements are not important.
2- Fracture mechanics of rubbers
The fracture properties of most materials at high strain can be roughly divided in two main processes: brittle fracture and ductile fracture. Brittle fracture occurs in the elastic region after a small deformation and high stress. When a sample loses its capacity to deform reversibly, the residual deformation after the sample is unloaded is called plastic deformation: polymer chains reorganize at large scale. If significant plastic deformation occurs at the tip of a propagating crack one talks about ductile fracture.
In the case of elastomers, the elastic modulus is low, but the material can sustain high levels of deformations almost reversibly before fracture. The deformation is quasi-elastic until fracture and the high increase in stiffness before crack propagation occurs is a sign of alignment of the polymer chains along the deformation direction. Unlike the case of glassy polymers, where plastic deformation is easily detectable by optical or electron microscopy, in rubbers, permanent damage is difficult to detect and few microscopic or molecular models exist. The most important of those is the 50 year-old Lake and Thomas model.
1 3 1 Eq. 47 2 3 Eq. 48 2 3 Eq. 49
2-1- Model of Lake and Thomas
We consider a crack propagating in a rubbery material along the x direction in a pure shear configuration as shown in Figure 6.
Figure 6: Scheme of a type I fracture under propagation2
We introduce the fracture toughness as the amount of energy dissipated per unit area created by the fracture. It is generally measured by the stored elastic energy el released by the
propagation of the crack of a unit area. In the case of a thin (plane stress) type I crack (i.e. loaded perpendicularly to the plane of propagation of the crack) propagation along the x direction, an expression of the fracture toughness was established.
with the thickness of the sample, E the modulus at small strain and the stress intensity factor.
It has indeed been demonstrated that ahead of the crack tip, the macroscopic stress applied to the sample was amplified locally. The stress amplification factor depends only on the elastic properties of the material and on test geometry.
We can now introduce 0 as the threshold value of below which the crack does not propagate
and from now will refer to 0 as the threshold fracture toughness.
Elastomers are polymer networks made of chains attached to each other by crosslinks. To propagate a crack, chains have to be broken somewhere. Lake and Thomas9 proposed a model to
account of the threshold fracture toughness of elastomers. 1
We consider a polymer chain crossing the plane of the crack as represented on Figure 7. This chain is made of monomers and has a fully extended length of , with the size of a monomer which is typically in the order of 2 Å.
Figure 7: Scheme of the crack propagation in a polymer network, polymer chain crossing the fracture plane The chain just ahead of the crack is highly extended and its stress/strain characteristics are described by a Langevin function.
At the point of rupture of the chain, the force applied is of the order of ~ ~ 4 , with the energy of a carbon-carbon bond (~ 4 eV) and the length of a carbon-carbon bond (~ 1.5 Å).
When the force increases, the polymer chain is increasingly extended and stores elastic energy
ent from an increase in entropy. Before the rupture force , the chain is extended and ent
saturates.
Then, the force applied to every links on the chains increases until one of those bonds breaks. At that point every bond is under . So when the chain breaks, the released energy is equal to the energy of a carbon-carbon bond times the number of bonds in the chain in addition to the entropic energy.
Near room temperature, is in the order of 1/40 eV and C C is in the order of a few eV. The stored energy of a chain just before breakage can be approximated at:
~ 1 2 3 3 2 Eq. 51 ~ C C ~ C C 3 2 Eq. 52 ~
To propagate, the crack needs to break every chain that crosses the plane of fracture. The fracture toughness predicted by Lake and Thomas can be written as Eq. 54.
with Σ the density of chains that cross the plane of fracture.
The probability to have a chain crossing the plane of fracture, Σ can be estimated for an ideal network of chains with monomers between crosslinks and leads to an approximation of the fracture toughness.
with C C ~ 2 and in the order of 1 Å, LT ~ 10 J.m-2.
This estimate of the fracture toughness LT from the model of Lake and Thomas is in good agreement with experimental values. It is furthermore predicting the evolution of with the molar mass of polymer chains as reported experimentally by Gent and Tobias10.
We may note that the Lake and Thomas approach does not consider the existence of a crack tip or the presence of a process zone or other dissipative mechanism ahead of the crack tip.
2-2- Experimental evaluation of the fracture toughness
The fracture mechanics approach is based on the quantification of the energy necessary to propagate a crack per unit area of crack and has been developed for small deformations and linear elastic materials. It is based on the determination of the value of the energy release rate from the elastic properties, test geometry and crack length, and equate this energy release rate with 0 when the crack starts to propagate.
For rubbers, several approaches have been developed to quantify the fracture toughness, or the fracture energy, as deterministic parameters. Thus, Rivlin and Thomas and later Greensmith11
proposed a rather simple method for the determination of the strain energy release rate in the case of single edge notched specimens:
LT~ Σ ~ Σ
C C Eq. 54
LT~ C C
2 Eq. 55
with W(λc) the strain energy density, λc the extension rate at break of the notched sample, c the
initial length of the crack and a strain-dependant empirical correction associated to the lateral contraction of the sample in extension.
The strain energy density W(λc) was calculated by integration of the stress strain curve of the un-notched sample, until λc.
Figure 8: Measure the strain energy density W(λc) in a single edge notched test , after measurement of λc
The strain dependence correction factor has been determined experimentally by Greensmith and is approximated by:
Hence, we used the following expression in this experimental part to determine the fracture energy at the onset of crack propagation. We determined λc at the maximum of stress for the
notched sample under uniaxial stretching.
3
Eq. 57
6cW
3- Network design
In this part are presented some strategies to design polymer networks and their incidence on the mechanical properties. We first focus on bimodal networks made by an end-linking reactions of various chains. Then results on interpenetrated networks are presented: elastomeric systems in the dry state and hydrogels swollen in water. Finally we report work on doubly crosslinked networks prepared under mechanical stress.
3-1- Bimodal networks
Bimodal networks are defined as polymer networks that contain two populations of chains lengths between crosslinks, typically a population of long chains and a population of short chains. They are prepared by end linking end functionalized chains creating hence a unique network. The key idea behind these bimodal networks is that short chains will reach their finite extensibility (and hence stiffen) for much lower levels of macroscopic extensions than the long chains.
Figure 9: Hypothetical sketch of a bimodal network, in which the short chains are arbitrarily drawn in a thicker line than the long chains12
The majority of the studies have been carried out on poly(dimethylsiloxane) (PDMS) or poly(urethane) due to an easy access to commercial end-functionalized polymer chains.
An extended study of the mechanical properties of PDMS bimodal elastomers has been reported by Mark et al.12,13. They worked on networks made of a blend of very short chains, with M
w
varying between 160 and 960 g.mol-1, and long chains with M
w ~ 18 500 g.mol-1. They showed
that for low concentrations of short chains, 10 – 20 mol %, ultimate properties were degraded by adding small chains. They argued that the deformation was manly supported by long chains, but that the fracture was breaking small chains preferentially. They also reported that with a very high molar content of short chains14 (70 to 90 mol %) the material starts to become tougher than
unimodal networks made of pure short or long chains as presented on Figure 10 taken from Llorente et al.14.
Figure 10: Stress/Strain curves for bimodal PDMS networks of various chain length and amounts of short chains (short 1100 – 660 or 220 g.mol-1 and long 18 500 g.mol-1), each curve is labeled with the molar percent of short chains14
For very high molar content of very small chains, the stress/strain curve shows a significant stiffening, signature of the finite extensibility of small chains that toughen the material. The maximum of λ is high, long chains having seemingly the ability to delay the fracture process. More recently Viers et al.15,16 and Cohen et al.17 showed that the bimodal elastomers may not be
homogeneous and display a segregation of small chains in more densely crosslinked clusters (Figure 11). They conclude that small chains had to be well dispersed in the polymer network to induce an enhancement of the mechanical properties. They showed that the best enhancement was observed when the small chains were close to their overlap concentration18.
Genesky et al.18 on PDMS bimodal networks and Cristiano et al.19 on poly(urethane) bimodal
networks, quantitatively show no significant improvement of the fracture toughness compared to unimodal networks as presented on Figure 12 with experiments on notched samples.
The introduction in the network of a third chain length in a network was reported by Genesky et al.18. Those trimodal networks are formed of 10 mol % of long chains (97 000 g.mol-1) and
various proportions of intermediate (8 500 g.mol-1) and short (800 g.mol-1). They present slightly
higher values of fracture toughness than unimodal and bimodal networks. Authors do not give any interpretation of that result. Materials nevertheless are still relatively brittle and the fracture toughness is at best increased by a factor of two.
Figure 12: Fracture energy vs. Mc, equivalent weight between crosslinks calculated from the initial modulus, for unimodal, bimodal and trimodal networks18
The bimodal network design may lead to elastomers with enhanced ultimate properties for undamaged samples but no significant improvement of the fracture toughness on notched samples. The main effect is observed on the crack initiation and not on the propagation. The presence of a large amount of short chains is needed to see any benefits.
3-2- Interpenetrated networks
An interpenetrated network (IPN) consists of two or more polymer networks, at least one of which is polymerized and/or crosslinked in the immediate presence of the other20. IPNs have
been studied since the early 1910s and are the topic of numerous scientific papers and patents. They found important applications in diverse technologies such as solar cells21,22, drug delivery23,
tissue engineering24–26, polymer actuator27,28. They can be gathered in two groups: the first is
composed of dry materials, made of pure polymers. The second consists on swollen networks with a large amount of solvent, they are called IPN gels or double network gels. If the solvent is
3-2-1- Double networks hydrogels
Synthesis and structure
Over the past ten years, the main breakthrough on IPN gels has been made by Gong et al.24, who
developed the concept of super tough gels with double networks hydrogels (DN hydrogels) composed of two interpenetrated polymer networks swollen by ~ 90 wt % water . They are usually synthesized via a two-step sequential free radical polymerization, starting from the synthesis of the highly crosslinked polyelectrolyte network by UV-polymerization, followed by the swelling of this first network with the second neutral monomer solution and finishing by a second UV-polymerization with a small amount of chemical crosslinker. The double network gel is then immersed in water to reach swelling equilibrium.
In their original form24, DN gels were made of a first minority network of highly stretched and
highly crosslinked polyelectrolyte (poly(2-acrylamido-2-methylpropanesulfonic acid) PAMPS), and a second network of loosely crosslinked neutral and entangled poly(acrylamide) (PAAm). Since this first report, a lot of successful polymer pairs have been used to create tough DN gels. The essential feature of DN gels is that they consist of two kinds of polymers with a strong contrasting structure, which can be summarized in three points29–31. First a contrast in chain
stiffness between the two networks: a stiff and brittle first network such as a highly swollen polyelectrolyte and a soft and ductile second network such as an unstretched neutral polymer. Second a contrast in network concentration, the second network must be 20 - 30 times more concentrated than the first network32. Third a contrast in crosslinker concentration, the first
network has to be tightly crosslinked while the second network must be made of very high molecular weight chains and very loosely crosslined or connected to the fist network.
Mechanical properties and toughening mechanism
Amongst the numerous interests of DN gels, we can underline their very low friction coefficient33,34 and their impressive toughness and stiffness29 compared to simple hydrogels which
exhibit usually very poor mechanical properties. As presented on Figure 13a, the mechanical properties of DN gels is not just a linear combination of the mechanical properties of individual simple gels, but there is a synergetic effect between the two networks. A necking phenomenon is sometimes observed in tough DN gels35,36, a plateau of stress appears in tensile tests. After the
necking phenomenon, the damaged gel is very soft and presents an initial modulus equal to one tenth of the fresh modulus.
Figure 13: a) Compressive nominal stress/strain curves for the PAMPS/PAAm DN gel, PAMPS gel and PAAm gel, b) Stress strain curves under uniaxail extesion for a DN gel showing a necking phenomenon29
Under cyclic deformation, DN gels show an important hysteresis under deformation during the first cycle only (Figure 13b and Figure 14a). It is accompanied by a sharp decrease of the initial elastic modulus. This phenomenon has been attributed to an internal bond breakage mechanism, chains of the first network are broken when the sample is stretched or compressed. Those chains are indeed already overstressed due to a high degree of swelling and are responsible for the high initial modulus of the fresh sample. They break easily under low strain and induce the softening even at relatively small strain (λ > 1.3 – 1.5). Clusters of the first partially broken network are supposed to play the role of physical crosslinker of the second network29, or as connected
breakable fillers. Small angle neutron scattering experiments reveals that clustering effect appearing under stretch37,38 with a periodicity near 1.7 µm in the structure of the stretched gel.
Figure 14: a)Tensile hysteresis loops of t-DN gels measures in tensile cyclic test and b) evolution of the initial elastic modulus during cyclic tensile experiement39
At that polymer volume fraction, usual gels are very brittle and exhibit a fracture energy9,40–42 of
the order of 10-1 – 100 J.m-2. Even if they are also made of ~ 90 wt % of water, DN gels are
extremely tough and present fracture energies of the order of 102 to 103 J.m-2, which is 100 to
1000 times larger than that of normal PAAm gels (100 J.m-2) or PAMPS gels (10-1 J.m-2) with
similar polymer concentrations to the DN gels30,31,35,43–45. This peculiar behavior is attributed to
the creation of a large process zone ahead of the crack tip where energy is dissipated by bond breaking of the highly stretched first network chains as presented on Figure 15. A damaged zone near the crack tip was observed after tearing DN gels using AFM or optical microscopy36,46.
Around the crack, a zone of ~100 µm presents a lower modulus and diffuses the light differently.
Figure 15: a)Schematic of the mechanical toughening mechanism in DN gels, bond breaking of the first network chains before the final macroscopic breakage of the material39, image of the crack tip observed after tearing36
Model for the fracture of DN hydrogels
Based on the necking phenomena, and the mechanical hysteresis on DN gels, Tanaka46 and
Brown47,48 have independently proposed a similar model for explaining the extremely high
threshold fracture energy for such diluted systems. They both assume that ahead of the crack tip, a damaged zone is created, defined by the yield conditions of the material.
In the model of Tanaka46, the yield point is supposed to be identical to the conditions observed
when the DN in showing necking, defining a stress threshold σc. He also assumes that there is a
sharp boundary between the damaged and the un-damaged material ahead of the crack. The damaged zone is only defined by its width . The damaged zone ahead of the crack tip is assumed to behave like a very soft and purely elastic material with an intrinsic fracture energy 0
Figure 16: a) Structure of crack assumed in the model proposed by Tanaka, process zone near the crack tip; b) Assumed stress/strain relations for the undamaged and damaged zone46.
To propagate a crack the stress on the crack tip must increase to σc to break chains of the first network (point A in Figure 16b) forming a Dugdale type soft damaged zone under a high stress σc (point C in Figure 16b). At that stress level, chains of the second network are highly extended.
If the material is virtually unloaded from that point to its initial state (segment CBO in Figure 16b), the released elastic energy is well below the work done to go to point C during the loading process. Tanaka proposed the following equation to predict the fracture energy DN of DN
hydrogels (Eq. 59).
where h is the characteristic size of the process zone and the energy dissipated due to irreversible damage of the first network, and U(σc) the elastic energy density (J.m-3) for the
uniform stretching of the second network until the critical stress σc.
The model of Brown47 is a more molecular model, Brown considered that the fracture process
occurs in two stages. The stiff undamaged DN (with a high modulus E1) is stretched until a stress
σa where chains of the first network progressively break and create a damaged zone ahead of the
crack tip with a low modulus E2. The estimate of σa used by Brown assumes that it is directly
proportional to the toughness of the first network alone, which leads to a prediction of a lower yield stress for more crosslinked first networks. Once the region ahead of the crack tip is damaged, the macroscopic crack needs to propagate by breaking the long chains of the second network loosely connected to the damaged second network. Brown assumes then that the
. . 1
damaged zone is a Dugdale thin strip and that the fracture criterion at the tip of the strip can be written as:
i.e: the strip breaks when the second network breaks. However just like for crazes the total is the work to stretch the strip from h0 to h at a stress of σa. Brown finally combines the equations
to obtain the fracture toughness of the DN hydrogel as given in Eq. 61.
Figure 17: a) Geometry of the damaged zone around the crack; b) Assumed stress/strain relation for the DN gel showing the yield-like situation cased by multiple cracking on the loading part of the cycle and the very low modulus of just the second network on unloading
with 2 the fracture toughnesses of the second network, λm the extension of the second network
leading to macroscopic crack, E2 the modulus of the second network alone and σ the stress at
which the first network breaks.
Both models predict the order of magnitude increase in fracture toughness of the DN hydrogels compared in comparison with the equivalent single networks hydrogels.
Recent developments on DN gel structure
Very recently, Nakajima et al.49 introduced the idea of molecular stent, small molecules or
polymers introduced in a neutral polymer network and used to increase the swelling ratio of a neutral network relative to the equilibrium of swelling. Combining molecular stent and DN gel structure, they prepared fully neutral tough DN hydrogels. They used poly(acrylamide) and
1
2 Eq. 60
4σ
poly(dimethyl acrylamide) as first and/or second polymers, as well as poly(hydroxyethyl acrylate). They showed that the DN concept was universal for the toughening of hydrogels regardless of the nature of the polymers and specific interactions between networks.
To investigate the effect of the very heterogeneous structure of the first network in the mechanical reinforcement of DN gels, Nakajima et al. prepared a very homogeneous neutral tetra-PEG (poly(ethylene glycol)) first network50,51 with a well-defined chain length between
crosslinks and a low level of defects in the network structure. Using the previous molecular stent strategy, they prepared a DN52 with a homogeneous first network structure and observed
qualitatively the same toughening mechanism than in conventional DN gels polymerized entirely by free radical polymerization.
Starting from the first double network structure and composition, several research groups have used the double networks concept to synthesize tough hydrogels from a variety of different polymers53. For example tough hydrogels have been prepared from poly(ethylene glycol) and
poly(acrylic acid)54–56, with a potential application in corneal implant applications. Poly(vinyl
alcohol) and poly(ethylene glycol)57 or cellulose58–60 based double networks have been developed
in order to be tough and bio compatible materials. For a more detailed review one may refer to recent reviews32,53,61.
One of the similarities between those double networks is the sacrificial bonds. Once broken, bonds of the sacrificial network cannot reform and the properties of the damaged material are modified compare to the fresh sample. On a recent paper, Sun et al.62 reported the synthesis of
covalently and ionically crosslinked double network gels. They are composed of a chemically crosslinked network (polyacrylamide) and a physically crosslinked network (alginate with ionic crosslinks through calcium ions Ca2+). Impressively tough and highly extensible, those gels
dissipate energy in the bulk through unzipping ionic bonds. Very interestingly mechanical properties are recovered after a recovery time needed for physical bonds to reform.
3-2-2- Interpenetrated elastomers
We consider now IPNs in the dry state, i.e. without solvents. Although conceptually these materials appear to be a logical extension of DN hydrogels, there is a significant difference: compared to IPN gels the solvent is no longer helping the mixing of polymers at the molecular
